\(\int e^{\frac {1}{4} \text {arctanh}(a x)} x^2 \, dx\) [126]

Optimal result

Integrand size = 14, antiderivative size = 500 \[ \int e^{\frac {1}{4} \text {arctanh}(a x)} x^2 \, dx=-\frac {11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac {(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac {x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}+\frac {11 \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt {2+\sqrt {2}}}\right )}{128 a^3}+\frac {11 \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt {2-\sqrt {2}}}\right )}{128 a^3}-\frac {11 \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt {2+\sqrt {2}}}\right )}{128 a^3}-\frac {11 \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt {2-\sqrt {2}}}\right )}{128 a^3}+\frac {11 \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x} \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}\right )}{128 a^3}+\frac {11 \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x} \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}\right )}{128 a^3} \] Output:

-11/32*(-a*x+1)^(7/8)*(a*x+1)^(1/8)/a^3-1/24*(-a*x+1)^(7/8)*(a*x+1)^(9/8)/ 
a^3-1/3*x*(-a*x+1)^(7/8)*(a*x+1)^(9/8)/a^2+11/128*(2+2^(1/2))^(1/2)*arctan 
(((2-2^(1/2))^(1/2)-2*(-a*x+1)^(1/8)/(a*x+1)^(1/8))/(2+2^(1/2))^(1/2))/a^3 
+11/128*(2-2^(1/2))^(1/2)*arctan(((2+2^(1/2))^(1/2)-2*(-a*x+1)^(1/8)/(a*x+ 
1)^(1/8))/(2-2^(1/2))^(1/2))/a^3-11/128*(2+2^(1/2))^(1/2)*arctan(((2-2^(1/ 
2))^(1/2)+2*(-a*x+1)^(1/8)/(a*x+1)^(1/8))/(2+2^(1/2))^(1/2))/a^3-11/128*(2 
-2^(1/2))^(1/2)*arctan(((2+2^(1/2))^(1/2)+2*(-a*x+1)^(1/8)/(a*x+1)^(1/8))/ 
(2-2^(1/2))^(1/2))/a^3+11/128*(2-2^(1/2))^(1/2)*arctanh((2-2^(1/2))^(1/2)* 
(-a*x+1)^(1/8)/(a*x+1)^(1/8)/(1+(-a*x+1)^(1/4)/(a*x+1)^(1/4)))/a^3+11/128* 
(2+2^(1/2))^(1/2)*arctanh((2+2^(1/2))^(1/2)*(-a*x+1)^(1/8)/(a*x+1)^(1/8)/( 
1+(-a*x+1)^(1/4)/(a*x+1)^(1/4)))/a^3
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.14 \[ \int e^{\frac {1}{4} \text {arctanh}(a x)} x^2 \, dx=-\frac {(1-a x)^{7/8} \left (7 \sqrt [8]{1+a x} \left (1+9 a x+8 a^2 x^2\right )+66 \sqrt [8]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{8},\frac {7}{8},\frac {15}{8},\frac {1}{2} (1-a x)\right )\right )}{168 a^3} \] Input:

Integrate[E^(ArcTanh[a*x]/4)*x^2,x]
 

Output:

-1/168*((1 - a*x)^(7/8)*(7*(1 + a*x)^(1/8)*(1 + 9*a*x + 8*a^2*x^2) + 66*2^ 
(1/8)*Hypergeometric2F1[-1/8, 7/8, 15/8, (1 - a*x)/2]))/a^3
 

Rubi [A] (warning: unable to verify)

Time = 1.44 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.31, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {6676, 101, 27, 90, 60, 73, 854, 828, 1442, 1483, 1142, 25, 1083, 217, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 e^{\frac {1}{4} \text {arctanh}(a x)} \, dx\)

\(\Big \downarrow \) 6676

\(\displaystyle \int \frac {x^2 \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}dx\)

\(\Big \downarrow \) 101

\(\displaystyle -\frac {\int -\frac {\sqrt [8]{a x+1} (a x+4)}{4 \sqrt [8]{1-a x}}dx}{3 a^2}-\frac {x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt [8]{a x+1} (a x+4)}{\sqrt [8]{1-a x}}dx}{12 a^2}-\frac {x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {33}{8} \int \frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}dx-\frac {(1-a x)^{7/8} (a x+1)^{9/8}}{2 a}}{12 a^2}-\frac {x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {33}{8} \left (\frac {1}{4} \int \frac {1}{\sqrt [8]{1-a x} (a x+1)^{7/8}}dx-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\right )-\frac {(1-a x)^{7/8} (a x+1)^{9/8}}{2 a}}{12 a^2}-\frac {x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {33}{8} \left (-\frac {2 \int \frac {(1-a x)^{3/4}}{(a x+1)^{7/8}}d\sqrt [8]{1-a x}}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\right )-\frac {(1-a x)^{7/8} (a x+1)^{9/8}}{2 a}}{12 a^2}-\frac {x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}\)

\(\Big \downarrow \) 854

\(\displaystyle \frac {\frac {33}{8} \left (-\frac {2 \int \frac {(1-a x)^{3/4}}{2-a x}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\right )-\frac {(1-a x)^{7/8} (a x+1)^{9/8}}{2 a}}{12 a^2}-\frac {x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}\)

\(\Big \downarrow \) 828

\(\displaystyle \frac {\frac {33}{8} \left (-\frac {2 \left (\frac {\int \frac {\sqrt {1-a x}}{\sqrt {1-a x}-\sqrt {2} \sqrt [4]{1-a x}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {1-a x}}{\sqrt {1-a x}+\sqrt {2} \sqrt [4]{1-a x}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}\right )}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\right )-\frac {(1-a x)^{7/8} (a x+1)^{9/8}}{2 a}}{12 a^2}-\frac {x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}\)

\(\Big \downarrow \) 1442

\(\displaystyle \frac {\frac {33}{8} \left (-\frac {2 \left (\frac {\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\int \frac {1-\sqrt {2} \sqrt [4]{1-a x}}{\sqrt {1-a x}-\sqrt {2} \sqrt [4]{1-a x}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\int \frac {\sqrt {2} \sqrt [4]{1-a x}+1}{\sqrt {1-a x}+\sqrt {2} \sqrt [4]{1-a x}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}\right )}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\right )-\frac {(1-a x)^{7/8} (a x+1)^{9/8}}{2 a}}{12 a^2}-\frac {x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}\)

\(\Big \downarrow \) 1483

\(\displaystyle \frac {\frac {33}{8} \left (-\frac {2 \left (\frac {-\frac {\int \frac {\sqrt {2+\sqrt {2}}-\frac {\left (1+\sqrt {2}\right ) \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\int \frac {\frac {\left (1+\sqrt {2}\right ) \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\int \frac {\sqrt {2-\sqrt {2}}-\frac {\left (1-\sqrt {2}\right ) \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\int \frac {\frac {\left (1-\sqrt {2}\right ) \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}\right )}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\right )-\frac {(1-a x)^{7/8} (a x+1)^{9/8}}{2 a}}{12 a^2}-\frac {x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {\frac {33}{8} \left (-\frac {2 \left (\frac {-\frac {-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{\sqrt [4]{1-a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\frac {1}{2} \left (1+\sqrt {2}\right ) \int -\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{\sqrt [4]{1-a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{\sqrt [4]{1-a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\frac {1}{2} \left (1-\sqrt {2}\right ) \int -\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{\sqrt [4]{1-a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}\right )}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\right )-\frac {(1-a x)^{7/8} (a x+1)^{9/8}}{2 a}}{12 a^2}-\frac {x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {33}{8} \left (-\frac {2 \left (\frac {-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{\sqrt [4]{1-a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{\sqrt [4]{1-a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{\sqrt [4]{1-a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{\sqrt [4]{1-a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}\right )}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\right )-\frac {(1-a x)^{7/8} (a x+1)^{9/8}}{2 a}}{12 a^2}-\frac {x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\frac {33}{8} \left (-\frac {2 \left (\frac {-\frac {\sqrt {2-\sqrt {2}} \int \frac {1}{-\sqrt [4]{1-a x}+\sqrt {2}-2}d\left (\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\sqrt {2+\sqrt {2}}\right )+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\sqrt {2-\sqrt {2}} \int \frac {1}{-\sqrt [4]{1-a x}+\sqrt {2}-2}d\left (\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}\right )+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\sqrt {2+\sqrt {2}} \int \frac {1}{-\sqrt [4]{1-a x}-\sqrt {2}-2}d\left (\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\sqrt {2-\sqrt {2}}\right )}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\sqrt {2+\sqrt {2}} \int \frac {1}{-\sqrt [4]{1-a x}-\sqrt {2}-2}d\left (\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}\right )}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\right )-\frac {(1-a x)^{7/8} (a x+1)^{9/8}}{2 a}}{12 a^2}-\frac {x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {33}{8} \left (-\frac {2 \left (\frac {-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\arctan \left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\arctan \left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}\right )}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\right )-\frac {(1-a x)^{7/8} (a x+1)^{9/8}}{2 a}}{12 a^2}-\frac {x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {33}{8} \left (-\frac {2 \left (\frac {-\frac {-\arctan \left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (\sqrt [4]{1-a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (\sqrt [4]{1-a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )-\arctan \left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (\sqrt [4]{1-a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{2 \sqrt {2-\sqrt {2}}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (\sqrt [4]{1-a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}\right )}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\right )-\frac {(1-a x)^{7/8} (a x+1)^{9/8}}{2 a}}{12 a^2}-\frac {x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}\)

Input:

Int[E^(ArcTanh[a*x]/4)*x^2,x]
 

Output:

-1/3*(x*(1 - a*x)^(7/8)*(1 + a*x)^(9/8))/a^2 + (-1/2*((1 - a*x)^(7/8)*(1 + 
 a*x)^(9/8))/a + (33*(-(((1 - a*x)^(7/8)*(1 + a*x)^(1/8))/a) - (2*(-1/2*(( 
1 - a*x)^(1/8)/(1 + a*x)^(1/8) - (ArcTan[(-Sqrt[2 - Sqrt[2]] + (2*(1 - a*x 
)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 + Sqrt[2]]] - ((1 - Sqrt[2])*Log[1 + (1 - 
 a*x)^(1/4) - (Sqrt[2 - Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/2)/(2* 
Sqrt[2 - Sqrt[2]]) - (ArcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - a*x)^(1/8))/(1 + 
 a*x)^(1/8))/Sqrt[2 + Sqrt[2]]] + ((1 - Sqrt[2])*Log[1 + (1 - a*x)^(1/4) + 
 (Sqrt[2 - Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/2)/(2*Sqrt[2 - Sqrt 
[2]]))/Sqrt[2] + ((1 - a*x)^(1/8)/(1 + a*x)^(1/8) - (-ArcTan[(-Sqrt[2 + Sq 
rt[2]] + (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 - Sqrt[2]]] - ((1 + S 
qrt[2])*Log[1 + (1 - a*x)^(1/4) - (Sqrt[2 + Sqrt[2]]*(1 - a*x)^(1/8))/(1 + 
 a*x)^(1/8)])/2)/(2*Sqrt[2 + Sqrt[2]]) - (-ArcTan[(Sqrt[2 + Sqrt[2]] + (2* 
(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 - Sqrt[2]]] + ((1 + Sqrt[2])*Log[ 
1 + (1 - a*x)^(1/4) + (Sqrt[2 + Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)] 
)/2)/(2*Sqrt[2 + Sqrt[2]]))/(2*Sqrt[2])))/a))/8)/(12*a^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 828

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[R 
t[a/b, 4]], s = Denominator[Rt[a/b, 4]]}, Simp[s^3/(2*Sqrt[2]*b*r)   Int[x^ 
(m - n/4)/(r^2 - Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x] - Simp[s^3/(2*S 
qrt[2]*b*r)   Int[x^(m - n/4)/(r^2 + Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x] 
, x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && 
 GtQ[a/b, 0]
 

rule 854

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 
 1)/n)   Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n 
)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 
2^(-1)] && IntegersQ[m, p + (m + 1)/n]
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1442

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[d^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), 
x] - Simp[d^4/(c*(m + 4*p + 1))   Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 
 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x 
] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2* 
p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1483

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r)   In 
t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r)   Int[(d*r 
 + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N 
eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
 

rule 6676

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x) 
^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, m, n}, x] &&  !Int 
egerQ[(n - 1)/2]
 

Maple [F]

\[\int {\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}^{\frac {1}{4}} x^{2}d x\]

Input:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)*x^2,x)
 

Output:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)*x^2,x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 467, normalized size of antiderivative = 0.93 \[ \int e^{\frac {1}{4} \text {arctanh}(a x)} x^2 \, dx =\text {Too large to display} \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)*x^2,x, algorithm="fricas")
 

Output:

-1/768*(-(33*I + 33)*sqrt(2)*a^3*(-1/a^24)^(1/8)*log((1/2*I + 1/2)*sqrt(2) 
*a^3*(-1/a^24)^(1/8) + (-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) + (33*I - 33 
)*sqrt(2)*a^3*(-1/a^24)^(1/8)*log(-(1/2*I - 1/2)*sqrt(2)*a^3*(-1/a^24)^(1/ 
8) + (-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) - (33*I - 33)*sqrt(2)*a^3*(-1/ 
a^24)^(1/8)*log((1/2*I - 1/2)*sqrt(2)*a^3*(-1/a^24)^(1/8) + (-sqrt(-a^2*x^ 
2 + 1)/(a*x - 1))^(1/4)) + (33*I + 33)*sqrt(2)*a^3*(-1/a^24)^(1/8)*log(-(1 
/2*I + 1/2)*sqrt(2)*a^3*(-1/a^24)^(1/8) + (-sqrt(-a^2*x^2 + 1)/(a*x - 1))^ 
(1/4)) - 66*a^3*(-1/a^24)^(1/8)*log(a^3*(-1/a^24)^(1/8) + (-sqrt(-a^2*x^2 
+ 1)/(a*x - 1))^(1/4)) - 66*I*a^3*(-1/a^24)^(1/8)*log(I*a^3*(-1/a^24)^(1/8 
) + (-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) + 66*I*a^3*(-1/a^24)^(1/8)*log( 
-I*a^3*(-1/a^24)^(1/8) + (-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) + 66*a^3*( 
-1/a^24)^(1/8)*log(-a^3*(-1/a^24)^(1/8) + (-sqrt(-a^2*x^2 + 1)/(a*x - 1))^ 
(1/4)) - 8*(32*a^3*x^3 + 4*a^2*x^2 + a*x - 37)*(-sqrt(-a^2*x^2 + 1)/(a*x - 
 1))^(1/4))/a^3
 

Sympy [F]

\[ \int e^{\frac {1}{4} \text {arctanh}(a x)} x^2 \, dx=\int x^{2} \sqrt [4]{\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}\, dx \] Input:

integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/4)*x**2,x)
 

Output:

Integral(x**2*((a*x + 1)/sqrt(-a**2*x**2 + 1))**(1/4), x)
 

Maxima [F]

\[ \int e^{\frac {1}{4} \text {arctanh}(a x)} x^2 \, dx=\int { x^{2} \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {1}{4}} \,d x } \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)*x^2,x, algorithm="maxima")
 

Output:

integrate(x^2*((a*x + 1)/sqrt(-a^2*x^2 + 1))^(1/4), x)
 

Giac [F]

\[ \int e^{\frac {1}{4} \text {arctanh}(a x)} x^2 \, dx=\int { x^{2} \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {1}{4}} \,d x } \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)*x^2,x, algorithm="giac")
 

Output:

integrate(x^2*((a*x + 1)/sqrt(-a^2*x^2 + 1))^(1/4), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{\frac {1}{4} \text {arctanh}(a x)} x^2 \, dx=\int x^2\,{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{1/4} \,d x \] Input:

int(x^2*((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/4),x)
 

Output:

int(x^2*((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/4), x)
 

Reduce [F]

\[ \int e^{\frac {1}{4} \text {arctanh}(a x)} x^2 \, dx=\int \frac {\left (a x +1\right )^{\frac {1}{4}} x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {1}{8}}}d x \] Input:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)*x^2,x)
 

Output:

int(((a*x + 1)**(1/4)*x**2)/( - a**2*x**2 + 1)**(1/8),x)